Here once more, the auction house can be finally entered and you can go inside to bid on items. Each item will basically allow you to raise your bid two times; if you opt not to on any of them, you won't get it.
Depending on the time or whatever -- it is more of a percentage thing I haven't gotten down yet -- you'll see different items or if you have already bought that item:
-- The Angel Wings simply make the wearer Float. Costs 10,000 Gil.
-- The Angel Ring puts Regen on the wearer. Costs 20,000 Gil.
-- The Golem Magicite gives you Golem. He can use Earthen Wall to protect you physically, and teaches Cura, Protect, and Stop. Costs 20,000 Gil.
-- The Zona Seeker Magicite hands over Zona Seeker. It uses Magic Shield to up your Magic Defense, and teachs Rasp, Osmose, and Shell. Costs 10,000 Gil.
-- There are other times when stuff, such as talking Chocobos, appear. You cannot buy them...
Buy what you want; the first two items can later be bought at shops, and the Espers are mostly defensive. I'd at least go for the Espers, definitely -- if you need money, run around the overworld for a while.
Now, return to the overworld and go to the town at the east end of the world, east of the Veldt ... Thamasa.